// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#include "../Eigen/SpecialFunctions"
#include "main.h"
#include <limits.h>

// Hack to allow "implicit" conversions from double to Scalar via comma-initialization.
template<typename Derived>
Eigen::CommaInitializer<Derived>
operator<<(Eigen::DenseBase<Derived>& dense, double v)
{
	return (dense << static_cast<typename Derived::Scalar>(v));
}

template<typename XprType>
Eigen::CommaInitializer<XprType>& operator,(Eigen::CommaInitializer<XprType>& ci, double v)
{
	return (ci, static_cast<typename XprType::Scalar>(v));
}

template<typename X, typename Y>
void
verify_component_wise(const X& x, const Y& y)
{
	for (Index i = 0; i < x.size(); ++i) {
		if ((numext::isfinite)(y(i)))
			VERIFY_IS_APPROX(x(i), y(i));
		else if ((numext::isnan)(y(i)))
			VERIFY((numext::isnan)(x(i)));
		else
			VERIFY_IS_EQUAL(x(i), y(i));
	}
}

template<typename ArrayType>
void
array_special_functions()
{
	using std::abs;
	using std::sqrt;
	typedef typename ArrayType::Scalar Scalar;
	typedef typename NumTraits<Scalar>::Real RealScalar;

	Scalar plusinf = std::numeric_limits<Scalar>::infinity();
	Scalar nan = std::numeric_limits<Scalar>::quiet_NaN();

	Index rows = internal::random<Index>(1, 30);
	Index cols = 1;

	// API
	{
		ArrayType m1 = ArrayType::Random(rows, cols);
#if EIGEN_HAS_C99_MATH
		VERIFY_IS_APPROX(m1.lgamma(), lgamma(m1));
		VERIFY_IS_APPROX(m1.digamma(), digamma(m1));
		VERIFY_IS_APPROX(m1.erf(), erf(m1));
		VERIFY_IS_APPROX(m1.erfc(), erfc(m1));
#endif // EIGEN_HAS_C99_MATH
	}

#if EIGEN_HAS_C99_MATH
	// check special functions (comparing against numpy implementation)
	if (!NumTraits<Scalar>::IsComplex) {

		{
			ArrayType m1 = ArrayType::Random(rows, cols);
			ArrayType m2 = ArrayType::Random(rows, cols);

			// Test various propreties of igamma & igammac.  These are normalized
			// gamma integrals where
			//   igammac(a, x) = Gamma(a, x) / Gamma(a)
			//   igamma(a, x) = gamma(a, x) / Gamma(a)
			// where Gamma and gamma are considered the standard unnormalized
			// upper and lower incomplete gamma functions, respectively.
			ArrayType a = m1.abs() + Scalar(2);
			ArrayType x = m2.abs() + Scalar(2);
			ArrayType zero = ArrayType::Zero(rows, cols);
			ArrayType one = ArrayType::Constant(rows, cols, Scalar(1.0));
			ArrayType a_m1 = a - one;
			ArrayType Gamma_a_x = Eigen::igammac(a, x) * a.lgamma().exp();
			ArrayType Gamma_a_m1_x = Eigen::igammac(a_m1, x) * a_m1.lgamma().exp();
			ArrayType gamma_a_x = Eigen::igamma(a, x) * a.lgamma().exp();
			ArrayType gamma_a_m1_x = Eigen::igamma(a_m1, x) * a_m1.lgamma().exp();

			// Gamma(a, 0) == Gamma(a)
			VERIFY_IS_APPROX(Eigen::igammac(a, zero), one);

			// Gamma(a, x) + gamma(a, x) == Gamma(a)
			VERIFY_IS_APPROX(Gamma_a_x + gamma_a_x, a.lgamma().exp());

			// Gamma(a, x) == (a - 1) * Gamma(a-1, x) + x^(a-1) * exp(-x)
			VERIFY_IS_APPROX(Gamma_a_x, (a - Scalar(1)) * Gamma_a_m1_x + x.pow(a - Scalar(1)) * (-x).exp());

			// gamma(a, x) == (a - 1) * gamma(a-1, x) - x^(a-1) * exp(-x)
			VERIFY_IS_APPROX(gamma_a_x, (a - Scalar(1)) * gamma_a_m1_x - x.pow(a - Scalar(1)) * (-x).exp());
		}
		{
			// Verify for large a and x that values are between 0 and 1.
			ArrayType m1 = ArrayType::Random(rows, cols);
			ArrayType m2 = ArrayType::Random(rows, cols);
			int max_exponent = std::numeric_limits<Scalar>::max_exponent10;
			ArrayType a = m1.abs() * Scalar(pow(10., max_exponent - 1));
			ArrayType x = m2.abs() * Scalar(pow(10., max_exponent - 1));
			for (int i = 0; i < a.size(); ++i) {
				Scalar igam = numext::igamma(a(i), x(i));
				VERIFY(0 <= igam);
				VERIFY(igam <= 1);
			}
		}

		{
			// Check exact values of igamma and igammac against a third party calculation.
			Scalar a_s[] = { Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5) };
			Scalar x_s[] = { Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5) };

			// location i*6+j corresponds to a_s[i], x_s[j].
			Scalar igamma_s[][6] = {
				{ Scalar(0.0), nan, nan, nan, nan, nan },
				{ Scalar(0.0),
				  Scalar(0.6321205588285578),
				  Scalar(0.7768698398515702),
				  Scalar(0.9816843611112658),
				  Scalar(9.999500016666262e-05),
				  Scalar(1.0) },
				{ Scalar(0.0),
				  Scalar(0.4275932955291202),
				  Scalar(0.608374823728911),
				  Scalar(0.9539882943107686),
				  Scalar(7.522076445089201e-07),
				  Scalar(1.0) },
				{ Scalar(0.0),
				  Scalar(0.01898815687615381),
				  Scalar(0.06564245437845008),
				  Scalar(0.5665298796332909),
				  Scalar(4.166333347221828e-18),
				  Scalar(1.0) },
				{ Scalar(0.0),
				  Scalar(0.9999780593618628),
				  Scalar(0.9999899967080838),
				  Scalar(0.9999996219837988),
				  Scalar(0.9991370418689945),
				  Scalar(1.0) },
				{ Scalar(0.0), Scalar(0.0), Scalar(0.0), Scalar(0.0), Scalar(0.0), Scalar(0.5042041932513908) }
			};
			Scalar igammac_s[][6] = {
				{ nan, nan, nan, nan, nan, nan },
				{ Scalar(1.0),
				  Scalar(0.36787944117144233),
				  Scalar(0.22313016014842982),
				  Scalar(0.018315638888734182),
				  Scalar(0.9999000049998333),
				  Scalar(0.0) },
				{ Scalar(1.0),
				  Scalar(0.5724067044708798),
				  Scalar(0.3916251762710878),
				  Scalar(0.04601170568923136),
				  Scalar(0.9999992477923555),
				  Scalar(0.0) },
				{ Scalar(1.0),
				  Scalar(0.9810118431238462),
				  Scalar(0.9343575456215499),
				  Scalar(0.4334701203667089),
				  Scalar(1.0),
				  Scalar(0.0) },
				{ Scalar(1.0),
				  Scalar(2.1940638138146658e-05),
				  Scalar(1.0003291916285e-05),
				  Scalar(3.7801620118431334e-07),
				  Scalar(0.0008629581310054535),
				  Scalar(0.0) },
				{ Scalar(1.0), Scalar(1.0), Scalar(1.0), Scalar(1.0), Scalar(1.0), Scalar(0.49579580674813944) }
			};

			for (int i = 0; i < 6; ++i) {
				for (int j = 0; j < 6; ++j) {
					if ((std::isnan)(igamma_s[i][j])) {
						VERIFY((std::isnan)(numext::igamma(a_s[i], x_s[j])));
					} else {
						VERIFY_IS_APPROX(numext::igamma(a_s[i], x_s[j]), igamma_s[i][j]);
					}

					if ((std::isnan)(igammac_s[i][j])) {
						VERIFY((std::isnan)(numext::igammac(a_s[i], x_s[j])));
					} else {
						VERIFY_IS_APPROX(numext::igammac(a_s[i], x_s[j]), igammac_s[i][j]);
					}
				}
			}
		}
	}
#endif // EIGEN_HAS_C99_MATH

	// Check the ndtri function against scipy.special.ndtri
	{
		ArrayType x(7), res(7), ref(7);
		x << 0.5, 0.2, 0.8, 0.9, 0.1, 0.99, 0.01;
		ref << 0., -0.8416212335729142, 0.8416212335729142, 1.2815515655446004, -1.2815515655446004, 2.3263478740408408,
			-2.3263478740408408;
		CALL_SUBTEST(verify_component_wise(ref, ref););
		CALL_SUBTEST(res = x.ndtri(); verify_component_wise(res, ref););
		CALL_SUBTEST(res = ndtri(x); verify_component_wise(res, ref););

		// ndtri(normal_cdf(x)) ~= x
		CALL_SUBTEST(ArrayType m1 = ArrayType::Random(32); using std::sqrt;

					 ArrayType cdf_val = (m1 / Scalar(sqrt(2.))).erf();
					 cdf_val = (cdf_val + Scalar(1)) / Scalar(2);
					 verify_component_wise(cdf_val.ndtri(), m1););
	}

	// Check the zeta function against scipy.special.zeta
	{
		ArrayType x(10), q(10), res(10), ref(10);
		x << 1.5, 4, 10.5, 10000.5, 3, 1, 0.9, 2, 3, 4;
		q << 2, 1.5, 3, 1.0001, -2.5, 1.2345, 1.2345, -1, -2, -3;
		ref << 1.61237534869, 0.234848505667, 1.03086757337e-5, 0.367879440865, 0.054102025820864097, plusinf, nan,
			plusinf, nan, plusinf;
		CALL_SUBTEST(verify_component_wise(ref, ref););
		CALL_SUBTEST(res = x.zeta(q); verify_component_wise(res, ref););
		CALL_SUBTEST(res = zeta(x, q); verify_component_wise(res, ref););
	}

	// digamma
	{
		ArrayType x(9), res(9), ref(9);
		x << 1, 1.5, 4, -10.5, 10000.5, 0, -1, -2, -3;
		ref << -0.5772156649015329, 0.03648997397857645, 1.2561176684318, 2.398239129535781, 9.210340372392849, nan,
			nan, nan, nan;
		CALL_SUBTEST(verify_component_wise(ref, ref););

		CALL_SUBTEST(res = x.digamma(); verify_component_wise(res, ref););
		CALL_SUBTEST(res = digamma(x); verify_component_wise(res, ref););
	}

#if EIGEN_HAS_C99_MATH
	{
		ArrayType n(16), x(16), res(16), ref(16);
		n << 1, 1, 1, 1.5, 17, 31, 28, 8, 42, 147, 170, -1, 0, 1, 2, 3;
		x << 2, 3, 25.5, 1.5, 4.7, 11.8, 17.7, 30.2, 15.8, 54.1, 64, -1, -2, -3, -4, -5;
		ref << 0.644934066848, 0.394934066848, 0.0399946696496, nan, 293.334565435, 0.445487887616, -2.47810300902e-07,
			-8.29668781082e-09, -0.434562276666, 0.567742190178, -0.0108615497927, nan, nan, plusinf, nan, plusinf;
		CALL_SUBTEST(verify_component_wise(ref, ref););

		if (sizeof(RealScalar) >= 8) { // double
			// Reason for commented line: http://eigen.tuxfamily.org/bz/show_bug.cgi?id=1232
			//       CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res, ref); );
			CALL_SUBTEST(res = polygamma(n, x); verify_component_wise(res, ref););
		} else {
			//       CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res.head(8), ref.head(8)); );
			CALL_SUBTEST(res = polygamma(n, x); verify_component_wise(res.head(8), ref.head(8)););
		}
	}
#endif

#if EIGEN_HAS_C99_MATH
	{
		// Inputs and ground truth generated with scipy via:
		//   a = np.logspace(-3, 3, 5) - 1e-3
		//   b = np.logspace(-3, 3, 5) - 1e-3
		//   x = np.linspace(-0.1, 1.1, 5)
		//   (full_a, full_b, full_x) = np.vectorize(lambda a, b, x: (a, b, x))(*np.ix_(a, b, x))
		//   full_a = full_a.flatten().tolist()  # same for full_b, full_x
		//   v = scipy.special.betainc(full_a, full_b, full_x).flatten().tolist()
		//
		// Note in Eigen, we call betainc with arguments in the order (x, a, b).
		ArrayType a(125);
		ArrayType b(125);
		ArrayType x(125);
		ArrayType v(125);
		ArrayType res(125);

		a << 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
			0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999,
			0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999,
			999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999,
			999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999;

		b << 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
			999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
			999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
			999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
			999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
			0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379,
			31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
			999.999, 999.999;

		x << -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
			-0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
			-0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
			-0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
			-0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
			-0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
			-0.1, 0.2, 0.5, 0.8, 1.1;

		v << nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
			nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, 0.47972119876364683, 0.5, 0.5202788012363533, nan, nan,
			0.9518683957740043, 0.9789663010413743, 0.9931729188073435, nan, nan, 0.999995949033062, 0.9999999999993698,
			0.9999999999999999, nan, nan, 0.9999999999999999, 0.9999999999999999, 0.9999999999999999, nan, nan, nan,
			nan, nan, nan, nan, 0.006827081192655869, 0.0210336989586256, 0.04813160422599567, nan, nan,
			0.20014344256217678, 0.5000000000000001, 0.7998565574378232, nan, nan, 0.9991401428435834,
			0.999999999698403, 0.9999999999999999, nan, nan, 0.9999999999999999, 0.9999999999999999, 0.9999999999999999,
			nan, nan, nan, nan, nan, nan, nan, 1.0646600232370887e-25, 6.301722877826246e-13, 4.050966937974938e-06,
			nan, nan, 7.864342668429763e-23, 3.015969667594166e-10, 0.0008598571564165444, nan, nan,
			6.031987710123844e-08, 0.5000000000000007, 0.9999999396801229, nan, nan, 0.9999999999999999,
			0.9999999999999999, 0.9999999999999999, nan, nan, nan, nan, nan, nan, nan, 0.0, 7.029920380986636e-306,
			2.2450728208591345e-101, nan, nan, 0.0, 9.275871147869727e-302, 1.2232913026152827e-97, nan, nan, 0.0,
			3.0891393081932924e-252, 2.9303043666183996e-60, nan, nan, 2.248913486879199e-196, 0.5000000000004947,
			0.9999999999999999, nan;

		CALL_SUBTEST(res = betainc(a, b, x); verify_component_wise(res, v););
	}

	// Test various properties of betainc
	{
		ArrayType m1 = ArrayType::Random(32);
		ArrayType m2 = ArrayType::Random(32);
		ArrayType m3 = ArrayType::Random(32);
		ArrayType one = ArrayType::Constant(32, Scalar(1.0));
		const Scalar eps = std::numeric_limits<Scalar>::epsilon();
		ArrayType a = (m1 * Scalar(4)).exp();
		ArrayType b = (m2 * Scalar(4)).exp();
		ArrayType x = m3.abs();

		// betainc(a, 1, x) == x**a
		CALL_SUBTEST(ArrayType test = betainc(a, one, x); ArrayType expected = x.pow(a);
					 verify_component_wise(test, expected););

		// betainc(1, b, x) == 1 - (1 - x)**b
		CALL_SUBTEST(ArrayType test = betainc(one, b, x); ArrayType expected = one - (one - x).pow(b);
					 verify_component_wise(test, expected););

		// betainc(a, b, x) == 1 - betainc(b, a, 1-x)
		CALL_SUBTEST(ArrayType test = betainc(a, b, x) + betainc(b, a, one - x); ArrayType expected = one;
					 verify_component_wise(test, expected););

		// betainc(a+1, b, x) = betainc(a, b, x) - x**a * (1 - x)**b / (a * beta(a, b))
		CALL_SUBTEST(
			ArrayType num = x.pow(a) * (one - x).pow(b);
			ArrayType denom = a * (a.lgamma() + b.lgamma() - (a + b).lgamma()).exp();
			// Add eps to rhs and lhs so that component-wise test doesn't result in
			// nans when both outputs are zeros.
			ArrayType expected = betainc(a, b, x) - num / denom + eps;
			ArrayType test = betainc(a + one, b, x) + eps;
			if (sizeof(Scalar) >= 8) { // double
				verify_component_wise(test, expected);
			} else {
				// Reason for limited test: http://eigen.tuxfamily.org/bz/show_bug.cgi?id=1232
				verify_component_wise(test.head(8), expected.head(8));
			});

		// betainc(a, b+1, x) = betainc(a, b, x) + x**a * (1 - x)**b / (b * beta(a, b))
		CALL_SUBTEST(
			// Add eps to rhs and lhs so that component-wise test doesn't result in
			// nans when both outputs are zeros.
			ArrayType num = x.pow(a) * (one - x).pow(b);
			ArrayType denom = b * (a.lgamma() + b.lgamma() - (a + b).lgamma()).exp();
			ArrayType expected = betainc(a, b, x) + num / denom + eps;
			ArrayType test = betainc(a, b + one, x) + eps;
			verify_component_wise(test, expected););
	}
#endif // EIGEN_HAS_C99_MATH

	/* Code to generate the data for the following two test cases.
	N = 5
	np.random.seed(3)

	a = np.logspace(-2, 3, 6)
	a = np.ravel(np.tile(np.reshape(a, [-1, 1]), [1, N]))
	x = np.random.gamma(a, 1.0)
	x = np.maximum(x, np.finfo(np.float32).tiny)

	def igamma(a, x):
	  return mpmath.gammainc(a, 0, x, regularized=True)

	def igamma_der_a(a, x):
	  res = mpmath.diff(lambda a_prime: igamma(a_prime, x), a)
	  return np.float64(res)

	def gamma_sample_der_alpha(a, x):
	  igamma_x = igamma(a, x)
	  def igammainv_of_igamma(a_prime):
		return mpmath.findroot(lambda x_prime: igamma(a_prime, x_prime) -
			igamma_x, x, solver='newton')
	  return np.float64(mpmath.diff(igammainv_of_igamma, a))

	v_igamma_der_a = np.vectorize(igamma_der_a)(a, x)
	v_gamma_sample_der_alpha = np.vectorize(gamma_sample_der_alpha)(a, x)
  */

#if EIGEN_HAS_C99_MATH
	// Test igamma_der_a
	{
		ArrayType a(30);
		ArrayType x(30);
		ArrayType res(30);
		ArrayType v(30);

		a << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0,
			10.0, 100.0, 100.0, 100.0, 100.0, 100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;

		x << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05, 1.17549435082e-38, 1.17549435082e-38,
			5.66572070696e-16, 0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06, 0.333412038288,
			1.18135687766, 0.580629033777, 0.170631439426, 0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
			10.5830172417, 10.5020942233, 92.8918587747, 95.003720371, 86.3715926467, 96.0330217672, 82.6389930677,
			968.702906754, 969.463546828, 1001.79726022, 955.047416547, 1044.27458568;

		v << -32.7256441441, -36.4394150514, -9.66467612263, -36.4394150514, -36.4394150514, -1.0891900302,
			-2.66351229645, -2.48666868596, -0.929700494428, -3.56327722764, -0.455320135314, -0.391437214323,
			-0.491352055991, -0.350454834292, -0.471773162921, -0.104084440522, -0.0723646747909, -0.0992828975532,
			-0.121638215446, -0.122619605294, -0.0317670267286, -0.0359974812869, -0.0154359225363, -0.0375775365921,
			-0.00794899153653, -0.00777303219211, -0.00796085782042, -0.0125850719397, -0.00455500206958,
			-0.00476436993148;

		CALL_SUBTEST(res = igamma_der_a(a, x); verify_component_wise(res, v););
	}

	// Test gamma_sample_der_alpha
	{
		ArrayType alpha(30);
		ArrayType sample(30);
		ArrayType res(30);
		ArrayType v(30);

		alpha << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0,
			10.0, 100.0, 100.0, 100.0, 100.0, 100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;

		sample << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05, 1.17549435082e-38, 1.17549435082e-38,
			5.66572070696e-16, 0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06, 0.333412038288,
			1.18135687766, 0.580629033777, 0.170631439426, 0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
			10.5830172417, 10.5020942233, 92.8918587747, 95.003720371, 86.3715926467, 96.0330217672, 82.6389930677,
			968.702906754, 969.463546828, 1001.79726022, 955.047416547, 1044.27458568;

		v << 7.42424742367e-23, 1.02004297287e-34, 0.0130155240738, 1.02004297287e-34, 1.02004297287e-34,
			1.96505168277e-13, 0.525575786243, 0.713903991771, 2.32077561808e-14, 0.000179348049886, 0.635500453302,
			1.27561284917, 0.878125852156, 0.41565819538, 1.03606488534, 0.885964824887, 1.16424049334, 1.10764479598,
			1.04590810812, 1.04193666963, 0.965193152414, 0.976217589464, 0.93008035061, 0.98153216096, 0.909196397698,
			0.98434963993, 0.984738050206, 1.00106492525, 0.97734200649, 1.02198794179;

		CALL_SUBTEST(res = gamma_sample_der_alpha(alpha, sample); verify_component_wise(res, v););
	}
#endif // EIGEN_HAS_C99_MATH
}

EIGEN_DECLARE_TEST(special_functions)
{
	CALL_SUBTEST_1(array_special_functions<ArrayXf>());
	CALL_SUBTEST_2(array_special_functions<ArrayXd>());
	// TODO(cantonios): half/bfloat16 don't have enough precision to reproduce results above.
	// CALL_SUBTEST_3(array_special_functions<ArrayX<Eigen::half>>());
	// CALL_SUBTEST_4(array_special_functions<ArrayX<Eigen::bfloat16>>());
}
